1. Field of the Invention
This invention relates to non-linear processing of digital signals, that is digital signal processing in which a non-linear operation is performed such as, for example, gamma correction in television or other display device applications.
2. Description of the Prior Art
In signal processing systems, a non-linear operation will cause harmonics of the input signal to be generated. For example, in television broadcasting applications, it is generally necessary to compensate for the non-linear characteristics of display devices such as cathode ray tubes (CRTs) by an appropriate inverse non-linear operation on the television signal so that the required display characteristics will be obtained. This is known as gamma correction, each type of display device having a particular gamma characteristic. The consequence of subjecting a signal to such a non-linear operation is that each frequency component of the signal will generate a sequence of harmonics. In digital systems, the generation of such harmonics can cause so-called alias components which can give rise to objectionable effects, such as various forms of interference. Since the alias components arise from the presence of harmonics above the Nyquist frequency limit, that is, half the value of the sampling frequency of the digital signal, one way of overcoming the problem is to design the system in such a way that any harmonics which are generated by the non-linear operation will always remain less than the Nyquist frequency limit; in practice this would mean substantially increasing the sampling frequency, and this results in a number of difficulties outlined in more detail below.
FIG. 1 of the accompanying drawings shows a typical non-linear characteristic of a gamma correction circuit for a television signal. A typical input/output characteristic of a television picture display device such as a CRT may be described by the following power law: EQU L=k.E.sup..gamma.
in which L is the light output of the CRT, k is a constant, E is the signal voltage applied to the CRT, and .gamma. (gamma) is a constant.
In any particular system, gamma will be a constant, and for CRTs, gamma will be between about 2.2 and 2.5.
The correction characteristic of FIG. 1 therefore needs to be applied in order to overcome the effect of the gamma characteristic, the signal voltage thereby being conditioned to be the inverse of the display device gamma characteristic. Due to the non-linear, and specifically logarithmic, nature of the inverse transformation between input and output amplitudes of the signal voltage, the frequency components of the input signal voltage generate harmonics, for example as shown in FIG. 2 of the accompanying drawings. In FIG. 2, a single (original) frequency component is shown as generating three harmonics at multiples of the original frequency, and in practice, all the frequency components of the signal will generate similar harmonics. Also, any non-linear operation on a signal will generate harmonics, the particular relative amplitudes of the harmonics depending on the specific nature of the non-linear operation.
In digital systems, as shown in FIG. 3 of the accompanying drawings, the harmonics which are generated by a non-linear operation can cause the above-mentioned undesirable alias components. In FIG. 3, an original frequency component of a digital signal sampled at a sampling frequency fs and subjected to a non-linear operation generates a second harmonic above the Nyquist limit fs/2. If the original frequency component has a frequency fo, the alias component fa will be disposed symmetrically about the Nyquist limit fs/2 with reference to the second harmonic, which will be at a frequency 2fo. Thus: EQU fs/2-fa=2fo-fs/2
and so: EQU fa=fs-2fo.
Such alias components can cause objectionable effects, for example, picture distortion in television applications.
One method of overcoming the problem of aliasing in non-linear digital processing would be to increase the sampling frequency such that any harmonics which are generated as a result of the non-linear operation always remain below the Nyquist frequency limit. FIG. 4 of the accompanying drawings illustrates such a situation in which the sampling frequency is substantially increased to fs' such that all significant harmonics of the original frequency component of the digital signal are less than the Nyquist frequency limit fs'/2. The increased sampling frequency could be achieved by super-sampling or up-conversion, that is, by interpolation. However, this apparent solution is unlikely to be practical in most circumstances, for the following reasons. Firstly, in a particular system the sampling frequency may need to be fixed and so up-conversion, particularly to the extent required to deal with a number of harmonics, will not be possible. Secondly, the high processing rate caused by substantially increasing the sampling frequency may be excessive in practice. Thirdly, it may be necessary subsequently to down-convert the processed digital signal to the original sampling frequency, or to filter back the signal to the original bandwidth. Either of these processes would remove the harmonics generated outside the bandwidth by the non-linear operation and would also therefore tend to negate the effect of the non-linear operation.